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%I #6 Jan 24 2013 14:53:16
%S 3,-16,4,88,-48,8,-496,432,-144,16,2848,-3456,1728,-384,32,-16576,
%T 25920,-17280,5760,-960,64,97408,-186624,155520,-69120,17280,-2304,
%U 128,-576256,1306368,-1306368,725760,-241920,48384,-5376,256,3424768,-8957952,10450944,-6967296,2903040,-774144,129024
%N A triangle of coefficients of a Moebius-transformed Pascal triangle as a sum: b(x,y,n)=Sum[Binomial[n,i]*x^i*y^(n-i),{i,0,n}]; transforms: x'->(a1*x + b1)/(c1*x + d1); y'->(a2*y + b2)/(c2*y + d2); b1(x,y,n)=(c1*x + b1)^(k)*(c2*y + d2)^(k)*b(x',y',n); f(x,y,z,n)=b1(x,y,n)+b1(y,z,n)+b1(z,x,n).
%C Row sums are:
%C {3, -12, 48, -192, 768, -3072, 12288, -49152, 196608, -786432, 3145728}.
%D P. J. Olver, Classical Invariant Theory, Cambridge Univ. Press, p. 242.
%D McKean and Moll, Elliptic Curves, Function Theory,Geometry, Arithmetic, Cambridge University Press, New York, 199, page 172
%F b(x,y,n)=Sum[Binomial[n,i]*x^i*y^(n-i),{i,0,n}]; transforms: x'->(a1*x + b1)/(c1*x + d1); y'->(a2*y + b2)/(c2*y + d2); b1(x,y,n)=(c1*x + b1)^(k)*(c2*y + d2)^(k)*b(x',y',n); f(x,y,z,n)=b1(x,y,n)+b1(y,z,n)+b1(z,x,n); Out_n,m=Coefficients(f(x,y,z,n)).
%e {3},
%e {-16,4},
%e {88, -48, 8},
%e {-496, 432, -144, 16},
%e {2848, -3456, 1728, -384, 32},
%e {-16576, 25920, -17280, 5760, -960,64},
%e {97408, -186624, 155520, -69120, 17280, -2304, 128},
%e {-576256, 1306368, -1306368, 725760, -241920, 48384, -5376, 256},
%e {3424768, -8957952,10450944, -6967296, 2903040, -774144, 129024, -12288, 512}, {-20417536, 60466176, -80621568, 62705664, -31352832, 10450944, -2322432, 331776, -27648, 1024},
%e {121980928, -403107840, 604661760, -537477120, 313528320, -125411328, 34836480, -6635520, 829440, -61440, 2048}
%t a1 = 1; b1 = -2; c1 = 0; d1 = 1; a2 = 0; b2 = 1; c2 = 1; d2 = -2; p[x_, y_, k_] = (c1*x + b1)^(k)*(c2*y + d2)^(k)*Sum[Binomial[k, i]*((a1*x + b1)/(c1*x + d1))^i*((a2*y + b2)/(c2*y + d2))^(k - i), {i, 0, k}]; f[x_, y_, z_, k_] = p[x, y, k] + p[y, z, k] + p[z, x, k]; Table[ExpandAll[f[x, y, z, k]], {k, 0, 10}]; a = Table[CoefficientList[f[x, y, z, k] /. y -> 1 /. z -> 1, x], {k, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[f[x, y, z, k] /. y -> 1 /. z -> 1, x]], {k, 0, 10}]
%K uned,tabf,sign
%O 1,1
%A _Roger L. Bagula_, Jun 14 2008
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